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Chapter 3: Problem 29

Given \(\log _{b} 3=1.099\) and \(\log _{b} 5=1.609\), find each value. $$ \log _{b} 75 $$

Short Answer

Expert verified
The value of \(\log_b 75\) is 4.317.

Step by step solution

01

Understand the Problem

We are given the logarithms of two numbers (3 and 5) with base \(b\). We need to find the logarithm of 75 with the same base \(b\).
02

Express 75 in Terms of 3 and 5

Notice that 75 can be expressed as a product of 3 and 5: \(75 = 3 \times 5^2\).
03

Apply Logarithmic Properties

Use the properties of logarithms: \(\log_b (xy) = \log_b x + \log_b y\) and \(\log_b (x^n) = n \log_b x\). Therefore, \(\log_b 75 = \log_b 3 + \log_b (5^2)\).
04

Simplify the Expression

According to the properties, \(\log_b (5^2) = 2 \log_b 5\). So, \(\log_b 75 = \log_b 3 + 2 \log_b 5\).
05

Substitute Known Values

Substitute the given logarithm values: \(\log_b 3 = 1.099\) and \(\log_b 5 = 1.609\). The equation becomes \(\log_b 75 = 1.099 + 2(1.609)\).
06

Calculate Final Result

Calculate \(\log_b 75\): \(1.099 + 2(1.609) = 1.099 + 3.218 = 4.317\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Properties
Logarithmic properties are fundamental rules that make working with logarithms much easier and more intuitive. Understanding these properties is crucial for solving logarithmic equations and problems:
  • Product Property: This property states that the logarithm of a product is the sum of the logarithms of the factors. It can be expressed as: \[ \log_b (xy) = \log_b x + \log_b y \]This property helps break down complex logarithmic expressions into simpler parts, making calculations more manageable.
  • Power Property: According to this property, the logarithm of a power is the exponent times the logarithm of the base. It is written as: \[ \log_b (x^n) = n \log_b x \]This is especially useful when dealing with exponents, letting you manipulate and simplify expressions involving powers.
  • Quotient Property: This states that the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator: \[ \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \]This property often assists in simplifying expressions when you're dividing two numbers inside a logarithm.
For example, in the exercise given, you applied the product and power properties to successfully find the logarithm of 75 by breaking it into the logarithms of its factors and powers.
Change of Base Formula
The change of base formula is a helpful tool for evaluating logarithms with any base when you have access to a different logarithmic base, such as base 10 or base \( e \) (natural logarithms). This formula allows you to rewrite a logarithm in terms of other bases often found on calculators:\[ \log_b x = \frac{\log_c x}{\log_c b} \]Using this formula, you can convert a logarithm from one base to another by using a calculator-friendly base. For example:
  • If you need \( \log_2 8 \) but only have base 10 (common logarithms) available, you could use:\[ \log_2 8 = \frac{\log_{10} 8}{\log_{10} 2} \]Calculators typically handle base 10 or natural logarithms, thus making calculations approachable and less daunting.
This formula greatly expands the types of logarithms you can handle, as it's often easier to compute using bases that your calculator readily provides. In practice, this universal applicability can simplify your computational work.
Applications of Logarithms
Logarithms are not just theoretical constructs but have important real-world applications. They crop up in many fields and disciplines, transforming complicated multiplicative processes into more manageable additive ones.
  • Science: In scientific fields, logarithms are used to measure phenomena that span large ranges of values. For instance, the Richter scale for earthquake intensity and the pH scale for acidity both use logarithms to simplify complex data.
  • Finance: Logarithms can calculate compound interest in finance, turning the multiplication of principal and interest rate factors over time into simpler addition problems.
  • Information Theory: In the realm of information and data, logarithms help define key concepts like entropy and information content, allowing for quantification of the unpredictability or information within a dataset.
Understanding these applications shows how logarithms simplify real-world multiplicative and exponential relationships, turning seemingly complex problems into easy-to-manage solutions.

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